Given a locally finite connected infinite graph G, let the interval [p min(G), p max(G)] be the smallest interval such that if p \u3e p max(G), then every 1-independent bond percolation model on G with bond probability p percolates, and for p \u3c p min(G) none does. We determine this interval for trees in terms of the branching number of the tree. We also give some general bounds for other graphs G, in particular for lattices. © 2012 Cambridge University Press